Answer to Question #177365 in Calculus for Moel Tariburu

Question #177365

Evaluate the integration by substitution ∫▒sin2θ/√(〖cos〗^2+16) dθ


1
Expert's answer
2021-04-29T17:21:57-0400

We have to find value of the integral


I = sin2θcos2θ+16dθ\int \dfrac{sin2 \theta}{\sqrt{cos^2{\theta}+16}}d\theta


Using the substitution,


sin2θ=2cosθsinθsin2\theta = 2cos\theta sin\theta

= 1cos2θ+162sinθcosθdθ\int- \dfrac{-1}{\sqrt{cos^2{\theta}+16}}2sin\theta cos\theta d\theta


Substitute u=cos2θ+16u = cos^2{\theta}+16

dudθ=2cosθsinθ\dfrac{du}{d\theta} = -2cos\theta sin\theta

Hence,

dθ=du2cosθsinθd\theta = -\dfrac{du}{2cos\theta sin\theta}

Hence integral will become


duu-\int \dfrac{du}{\sqrt{u}}

=2u= -2\sqrt{u}

Using substitution u=cos2θ+16u = cos^2{\theta}+16


We get , I=2cos2θ+16+CI = -2\sqrt{cos^2{\theta}+16} + C



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